Hydrogen Plasma —Phase Diagram and Properties

W. Ebeling, Berlin; W.D. Kraeft, Greifswald and D. Kremp, Rostock(Humboldt-Universität, Ernst-Moritz-Arndt-Universität and Wilhelm-Pieck-Universität)

Hydrogen is not only the most simple element and the most abundant but also in some respects the most exotic. Here we are reporting on just the fluid plasma phases of hydrogen including the liquid metallic phase. Of special interest is the possibility of macroscopic quantum ef­fects as e.g. superfluidity and supercon­ductance.

The simplest hydrogen system con­sisting of one electron and one proton only, comprises bound states of the atom:

e- + p+ ↔ Hwhile a four particle system comprises atomic, ionic as well as molecular sta­tes:

↔ 2H

In other words we have different reac­tion channels. In many particle systems all the reactions mentioned above may occur, modified and influenced by many particle, temperature or pressure ef­fects 1). The manifold of possible behav­iours of hydrogen systems as a function of temperature, pressure or kinetic con­ditions is thus quite extensive 2).

The existence of a metallic state of hydrogen matter was first postulated by Wigner and Huntington in 1935. Since then, many authors 3-5) have discussed the properties of liquid metallic hydro­gen including the possibility of macro­scopic quantum effects, as e.g. super-

Fig. 1 — Phase diagram of hydrogen and the adiabate of Jupiter (dash-dotted) and Sun (dashed).

conductance. We should mention also the interesting experimental investiga­tions in which the conditions close to the required temperatures and densities have been approached 3). However, one must say that both the experimental and theoretical work in this field are still far from providing solutions to the many problems that are open. One of the most important questions is the metastability of high pressure phases, with the theo­retical possibility of having metastable phases of hydrogen with unusual physi­cal properties even at normal pressures and temperatures. This could of course have very important technical applica­tions as e.g. high-temperature supercon­ductors.Fluid Phases of Hydrogen

In spite of the simplicity in composi­tion, hydrogen systems have a great variety of possible stable phases. The phase diagram of hydrogen, which is still hypothetical in the high-pressure region, is shown in Fig. 1. Let us consider some isobars and isotherms, beginning with the isobar p = 103 bar (108 Pa). Coming from the high-temperature side, we are first in the plasma region; then at about 104 K we come to the narrow region of atomic hydrogen which, below 103 K, gives place to a molecular region. Finally, with decreasing temperature the hydro­gen crystallizes and forms a H2 solid. At pressures below

Pc = 1.3 MPa (Tc = 33 K) a line of coexistence between a mo­lecular liquid and gas phase is obser­ved. In other words at a pressure of, e.g. p = 105 Pa (1 bar), the system moves with decreasing temperature through a state with two coexisting fluid phases. Below the triple point the hydrogen sy­stem goes directly from the H2 gas phase to the H2 solid phase.

Let us now consider an isotherm at say T = 1 K. Starting with low pres­sures, the system is in the H2 solid state. With increasing pressure the solid breaks down at about p = 1011 -1012 Pa (1-10 Mbar) and goes to the hypothetical metallic hydrogen solid. The precise transition pressure is not known. How­ever several estimates yield values in the Megabar region. For isotherms between

Tt = 1700 K (pt = 230 GPa), which is the estimate for the second

triple point of hydrogen after Kerley (1972), and

Tc = 16500 K (pc = 23 GPa), which is the estimate for the second critical point of hydrogen 2), (see also ref. 4), another (hypothetical) phase transition is expected. This is the so- called plasma phase transition discuss­ed first by Norman and Starostin (1968), and by Ebeling, Kraeft and Kremp (1976). Along the coexistence line, the plasma splits into two phases with dif­ferent degrees of ionization: a highly ionized phase and a weakly ionized phase, see below.Fig. 2 — Physical states in the density — temperature diagram (dotted line: Ae=d; dash-dotted: l=d).

Another approach to discussing the phases of hydrogen makes reference to the density-temperature plane. Here it is useful to start from a consideration of the characteristic lengths of the system — see panel.

ae = 0.5292x10-10 m - electronic Bohr radius,

ap = 0.2882x10-15 m - protonic Bohr radius

d = (3/4/πnp)1/3 -mean distance between protons

Ae = h (2πnmekBT)-1/2 - thermal De Broglie wave length of electrons

Ap - corresponding quantity for protons

I = e2/ kBT - Landau lengthThe border of strong coupling is givenby:

I s dand the borders of degeneracy by

Ae = d and Ap s d.

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Universiteit van Amsterdam

The University of Amsterdam has an opening for a

professor of condensed matter physics (m/f)*Full-time.Vacancynumber E.N. 0823.

The appointed candidate’s research program will constitute part of the research program of the Natuurkundig Laboratorium. This laboratory is involved in a broad spectrum of research projects in condensed matter physics.The candidate should lead a vigorous research program and should be able to obtain funding from external science­supporting agencies.

Areas of research which have priority are metal physics, magnetism and semiconductor physics. It is desirable that the research program of the appointed candidate is linked to modem developments in material science. The Natuurkundig Laboratorium has experimental facilities for performing experiments at low temperatures, at high magnetic fields and at high pressures. Furthermore there is equipment to measure magnetic, thermal and optical properties, and there is an extended facility for growing single (metal) crystals.

Gross salary depends on age and experience and ranges from Dfl. 6,854.- to Dfl. 9,495.- per month (Dutch Civil Servants Code).

Applicants are requested, quoting vacancynumber, to send a curriculum vitae, list of publications and the names and addresses of a few referees, willing to provide information concerning personal and scientific qualifications - within four weeks after date of publication - to Prof. Dr. A. Lagendijk, Natuurkundig Laboratorium, Valckenierstraat 65,1018 XE Amsterdam, The Netherlands.

Information concerning the position can be obtained from the Chairman of the Search Committee Prof. Dr. A. Lagendijk, telephone 020-522 2716.

Those wishing to draw attention to potential candidates are welcome to contact the Committee Chairman.

* Women in particular are invited to apply.

At d < 102 / the protons may form a Wigner lattice. At d < ap the protons behave like an ideal Fermi fluid, and at d < ae the electrons may be considered to be ideal. Different regions correspon­ding to this classification are shown in Fig. 2.

Understanding the physical properties of many particle hydrogen systems is not just of inherent interest but also of practical significance. In the first place there are direct astrophysical applica­tions 3,4). With a level of about 70%, hydrogen is the most abundant of the elements and to a certain approximation the early stages of the evolution of our Universe from an age of say 10 minutes to 1 million years corresponded to a hot hydrogen-helium plasma which was adiabatically expanding. In other words, the Universe moved along an adiabate, crossing at about 105 years after the big bang the conditions for the formation of bound states (atoms and molecules). The interior of stars and the giant planets covers a wide range of states as can be seen in Fig. 1 where two adia- bates are plotted, corresponding to the states along a radial section of Jupiter and the Sun. In the centre of the giant planets are conditions (pressures up to 35x1016 Pa and temperatures up to 20000 K) which are in the region of a liquid metallic state 4).

Effective Interaction and Bound States 1·6)

From the point of view of quantum mechanics, the behaviour of an isolated e-p pair is described by the Schrödinger equation, the solutions of which are well known. For positive energies we have a scattering state solution, i.e., the pro­bability is finite even for very large separation of the electron and the pro­ton. For the bound state solutions, which correspond to negative energy ei­genvalues, the probability is zero to find the proton very far from the electron (H- atom). On adding certain energy, the bound H-atom may be transferred into a scattering state (ionization) as shown in Fig. 3.

Fig. 3 — The density dependence of the continuum edge and of the bound states.

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Fig. 4 — Feynman diagrams.

Entirely new aspects must be con­sidered if the e-p pair interacts with the surrounding medium (plasma) and the problem becomes rather complicated. The behaviour of the H-atom is modified by three effects:

The first is the self energy, which takes into account the modifications arising from the propagation of a single e or p through the plasma. This is illustra­ted in Fig. 4 where the propagation of a free particle is represented by an arrow (Fig. 4a), and that of an interacting one by a fat arrow (Fig. 4b). This may be ex­pressed by the relation of Fig. 4c where the interaction of the e(p) with the plasma is represented by the self energy

Fig. 5 — The equation of state in the high pressure region. Theoretical estimates 2) for the metallic liquid in comparison with se­veral points obtained from adiabatic com­pression experiments 3).

Fig. 6 — Degree of ionization as a function of the density.

Σ. Obviously the behaviour of a pair is in­fluenced by the self energy and we may say that we have to deal with the bound state problems of quasi particles, which are, separately, at zero momentum in an energy valley (E < 0).

A second typical effect is the screen­ing of the long range Coulomb potential by the surrounding plasma, so that in reality, the e-p pair in a plasma interacts via a screened potential, which in the language of Feynman diagrams may be represented as in Fig. 4d. The screened potential is thus essentially determined by the polarization π of the plasma.

The last essential effect is the so- called Pauli-blocking, according to which the interaction between e and p is possible only if the final states are not occupied. Symbolically this may be re­presented as in Fig. 4e where ne(p1) is the Fermi distribution. Taking into ac­count the three processes mentioned above, we may derive an effective Schrödinger equation which describes the behaviour of the e-p pair, and espe­cially the modification of the properties of the H-atom, by the interaction with the plasma. It takes into account many body effects through density and tem­perature-dependent terms. For example the potential appears always in connec­tion with «blocking factors» containing the Fermi distribution.

An approximation of physical interest is the case in which the effective interac­tion describes the decay of an H-atom into a free e-p pair under stimulated emission of a plasmon with frequency ωpl and the inverse process, in which a plasmon is absorbed, and the final state is again a free one (Fig. 4f).

If we consider the correction terms in the effective Schrödinger equation to be a perturbation, the energy eigenvalues of the e-p pair in the plasma may be de­termined. The results are represented in Fig. 3. We see that the ground state is only weakly density dependent whereas in contrast, the continuum edge, i.e. the energy of scattering states for p1=p2= 0, is strongly density depen­dent on account of the density depen­dence of the self energy.

Obviously the difference between the ground state energy and the continuum edge is the ionization energy. From Fig. 3 it follows that this decreases with in­creasing density, the cross-over bet­ween the ground state energy and the continuum edge defining the Mott den- sity V nMott. At higher densities, n > nMott, bound states do not exist, an effect that turns out to be essential for understan­ding the behaviour of matter at high pressures, and it determines crucially the thermodynamic and transport pro­perties of H-plasmas.Equation of State and Ionization Equilibrium

Fully ionized fluid phases may be con­sidered to be a mixture of two Fermi fluids. Fig. 5shows the result of theoreti­cal estimates for the high pressure re­gion of the equation of state 2) in com­parison with experimental results 3). The highest pressures reached are still below 1012 Pa and the results are loaded with uncertainties. One may hope that new experiments which are under way will shed new light on this question.

For regions with bound states, the thermodynamic description is much more complicated. In this case, e.g., the pressure is determined by all contribu­tions of elementary and composite par­ticles and the scattering between them. Restricting our considerations to fluid phases only, the equation of state must cover the partially ionized plasma and the neutral liquid and gas as well. Such equations were considered in 1,2,4). In addition one must use an equation for the plasma composition, i.e., a mass action law, or Saha equation.

For the single reaction : e- + p+ ↔ H

the balance equation for the densities reads:

np = ne = ne* + nh, n* = n*where the stars denote the densities of free species. Chemical equilibrium con­ditions yield the Saha equation

(Aep = h (2π(me + mp)kBT)-1/2, thermal wavelength). Corresponding equations are valid for other reactions such as:

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H + H ↔ H2.Mass action laws give the composi­

tion of the partially ionized plasma, while the effective ionization energy, /eff, must be determined from statistical theory 1·2).

From the discussion of the bound state problem and the relevant effective interaction we know that at a certain density all bound states vanish. This Mott effect for the two particle bound state energies leads to a steep increase in the degree of ionization, i.e., it chan­ges the composition of the plasma drastically. This effect is referred to as the Mott transition and may be seen in Fig. 6, where we show the degree of ionization as a function of the density for different temperatures.

With an equation of state and the plasma composition we may determine the pressure or other thermodynamic quantities as a function of density and temperature for a partially ionized plasma.

Plasma Phase Transition 1·2)Of high interest in the question of

phase transitions, as noted above. One has, e.g., to investigate the violation of the stability conditions dp / dV < 0 which signals the possibility of phase transitions of first order. Alternatively we may consider the chemical potential of the plasma, p, when the correspon­ding stability condition is:

dp / ∂n > 0 , ne = Ne IV.Fig. 7 shows an example, where we

find a violation of the stability condition. We observe a typical van-der-Waals-like behaviour, signalling the possibility of a phase transition of first order. This phase transition is the plasma phase transition discussed at the beginning. In the vicinity of the critical point it is essentially determined by the Coulomb interaction and the ionization equili­brium. At lower temperatures the inter­action between neutral particles is im­portant.

Fig. 7 — Chemical potential as a function of the density.

Fig. 8 — The line of coexistence for the plasma phase transition. A - fluid molecular H B - weakly ionized H C - strongly ionized H t2 - estimate of the second

triple pointC2, C'2 — estimate of the

second critical point in two approximations

D - liquid metal E - molecular solid.

The main feature of the phase transi­tion is the coexistence of two phases having different degrees of ionization. The low density phase is a partially ioniz­ed plasma reaching from a highly ionized plasma to a molecular fluid. The high density phase is the metallic liquid.

The corresponding coexistence curve is indicated in Fig. 1 and is shown in more detail in Fig. 8. A phase transition of such a type is predicted by the theory and may be referred to as a Mott phase transition although experimentally such a transition has (at least for hydrogen), not been detected yet; experiments 3) do not cover the relevant area comple­tely.

REFERENCES1. Kraeft W.D. et al., Quantum Statistics of Charged Particles, (Akademie-Verlag, Berlin) 1985; (Plenum Press, N.Y.) 1985.2. Schlanges M. and Kremp D., Ann. Physik 39 (1982) 69; Phys. Lett. 110A (1985); Ebel- ing W. and Richert W., Phys. Lett. 108A (1985) 80; Physica 130A (1985) 587.3. Grigoriev F.V. et al. Sov. Phys JETP 69 (1975) 743; Nellis A. et at., Phys. Rev. A 27 (1983) 608.4. Robnik M. and Kundt W., Astron. Astro- phys. 120 (1983) 227.5. Carlson A.E. and Ashcroft N.W., Phys. Rev. B 29 (1984) 479; Campbell C.E. and Zabolitzky J.G., Phys. Rev. B 29 (1984) 123.6. Zimmermann R., Kilimann K., Kraeft W.D., Kremp D. and Röpke G., Phys. Stat. Sol. (b) 90 (1978) 175.

Kernfysisch Versneller Instituut, Groningen The Netherlands

Staff Position in

Experimental Nuclear PhysicsThe KVI wishes to appoint a young experimental nuclear physicist on a tenured staff posi­tion. The appointment will be with FOM, the Netherlands' Foundation for Fundamental Research on Matter.The Kernfysisch Versneller Instituut is a national research institute in nuclear physics, jointly sponsored by the University of Groningen and by FOM. It has a broad experimental and theoretical research program on nuclear structure and heavy-ion physics.At present the main facility of the KVI is a K=160 MeV AVF-cyclotron for the acceleration of light and heavy ions. A large variety of detectors such as a magnetic spectrograph and various systems for the detection of light and heavy ions, electrons and y-radiation is availa­ble. Recently funds have been made available for the design and construction of a K=600 MeV superconducting cyclotron for light and heavy ions. The machine which will be built in Orsay in close collaboration with the Institut de Physique Nucléaire, will replace the present K=160 MeV cyclotron at the KVI in 1991.The successful candidate should be interested in the research that can be performed with the existing and future facility, and should also be willing to take on responsibilities for new instrumental developments. Several years of post-doctoral experience are required.Applicants for the above-mentioned position are requested to submit a curriculum vitae, list of publications and the names and addresses of three references, as early as possible but not later than 15 May 1986, to: Prof dr R H. Siemssen

Kernfysisch Versneller Instituut, Zernikelaan 25 NL - 9747 AA Groningen, The Netherlands

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